<.'i^_\,^'i'T;>j 


^ii^M 


Ube  XHniversitp  ot  Cbicaao 


SOME  SPECIAL  CASES  OF  THE  FLECNODE 

TRANSFORMATION  OF  RULED 

SURFACES 


A  DISSERTATION 

SUBMITTED   TO   THE   FACULTY 

OF  THE   ODGEN   GRADUATE   SCHOOL   OF  SCIENCE 

IN   CANDIDACY  FOR  THE  DEGREE    OF 

DOCTOR   OF  PHILOSOPHY 

DEPARTMENT   OF  MATHEMATICS 


BY 


JOHN  WAYNE  LASLEY,  JR. 


Private  Edition,  Distributed  By 

THE  UNIVERSITY  OF  CHICAGO  LIBRARIES 

CHICAGO,  ILLINOIS 

1922 


XTbe  TaniverBiti?  of  CblcaQo 


SOME  SPECIAL  CASES  OF  THE  FLECNODE 

TRANSFORMATION  OF  RULED 

SURFACES 


A  DISSERTATION 

SUBMITTED   TO   THE   FACULTY 

OF  THE   ODGEN   GRADUATE   SCHOOL  OF  SCIENCE 

IN  CANDIDACY  FOR  THE  DEGREE   OF 

DOCTOR   OF  PHILOSOPHY 

DEPARTMENT   OF   MATHEMATICS 


BY 

JOHN  WAYNE  LASLEY,  JR. 


Tl 


.O  . 


Private  Edition,  Distributed  By 

THE  UNIVERSITY  OF  CHICAGO  LIBRARIES 

CHICAGO,  ILLINOIS 

1922 


■••  t- 


Li 


INTRODUCTION 

The  general  theory  of  non-developable  ruled  surfaces  may  be 
made  to  depend^  upon  a  system 


(i) 


y"-\-piiy'-\rpi2z'+qiiy+qi2Z  =  o, 
2"  -\-p2iy'-\-p22z'-\-q2iy+q22Z  =  o 


of  two  second  order  linear  homogeneous  differential  equations, 
where  the  coefficients  pik,  qik  are  functions  of  x,  and  where  the 
strokes  indicate  differentiation  as  to  x.  According  to  the  funda- 
mental theory  of  differential  equations  these  equations  have  a 
general  solution,  a  pair  of  functions  y  and  z  of  x,  which  can  be 
expressed  as  linear  combinations'  with  constant  coefficients  of  the 
four  particular  solutions  {ji,  z^,  {{  =  1,2,2,,  a)  for  which  the  determi- 
nant 

y'l   y'2 

z[     z', 

yi   y^ 

Z\       Z2 


(2) 


D  = 


y\ 

z\ 

ys 

Zt 


does  not  equal  to  zero. 

If  we  interpret  ji  and  z,-  (i  =  i,  2,  3,  4)  as  the  homogeneous 
co-ordinates  of  two  points  in  space,  we  obtain  two  curves 

yk=yk{x),      Zk=zk(x)       (^=1,2,3,4). 

The  line  joining  those  points  which  correspond  to  the  same  values 
of  X  generates  a  ruled  surface.     This  ruled  surface  will  not  be  a 
developable  as  a  result  of  the  hypothesis  that  D  is  not  equal  to  zero. 
Let  us  transform  the  system  (i)  by  putting 

(3)  ri  =  ay-\-0z,  ^  =  yy-\-8z,  ^  =  ^{x), 

where  a,  /3,  7,  8,  ^  are  arbitrary  functions  of  x  for  which 

'  E.  J.  Wilczynski,  Projective  Diferential  Geometry  of  Curves  and  Ruled  Surfaces, 
Leipzig,  1906,  pp.  126  ff.,  hereafter  referred  to  as  W. 


492860 


FLECNODE  TRANSFORMATION  OF  RULED  SURFACES 


This  transforms  (i)  into  a  new  system  of  the  same  kind.  The  inte- 
grating ruled  surfaces  of  the  two  systems  are  the  same,  but  they 
are  referred  to  a  different  pair  of  directrix  curves  and  a  different 
independent  variable. 

The  fundamental  invariants  of  (i)  are^ 


(4) 


where 


(5)                     /■  =  7^„+M3 

and 

, 

Mi 

Mi 

(6) 

M, 

M, 

, 

Vi 

Vi 

(7) 

V2 

v^ 

f 

w 

w 

(8) 

W: 

W- 

6,  =P-4/,     ^4-i  =  8&4^;'-9^;'+8/^. 

e,,={p-^){K-r^)-\-{ir-2jy, 

till  M22  W12  W31 

Vij  —V22       fij         V21 

Wii—U'22         W'l2         U21 


^  =  Wli  +  M2J,         J  =  UiiU22,  —  Ui2U2l,         K  =  Vj{U22  —  Vi2V21, 


=  2pu  —  4qii-\-p\i-\-pi2p2i , 

=  2/>i2  — 49i2-f /'l2(/'li  +  /'22)    , 

^2Pji—4q2i-\-p2i{pii-\-p22)  , 

=  2p22  —  4q22-{-Pl2-i-pi2P2l  , 

■■  2Mii+^i2«2i  — />2,M,2, 
2Ui2-\-{pll-p22)Ul2-pl2iUll  —  U22), 

■■  2Mji—  (/>„  —  /) J ««  +  />„ (Mii  —  M J  , 

•  2Mj2         pI2'^2ll     p2l'^^I2  > 
=  2V'ji-\-Pl2V2t  —  P2lVl2, 

=  2f  1',+  (pii  -  />«)»«  - piiiVii  -  f 2j)  , 

=  2V2i—{pii—p22)V2l-\-p2l(Vll  —  V22)   , 
=  2V'22  —  pl2'V2i  +  p2lVl2- 

A  tangent  plane  intersects  a  surface  in  a  plane  curve  which  has  a 
double  point  at  the  point  of  contact.  If  one  of  the  double-point, 
tangents  is  an  inflectional  tangent  to  this  plane  curve,  the  point 
of  contact  is  called  a  flecnode,  a  name  due  to  Cayley.'  From  the 
point  of  view  of  Salmon,^  followed  here,  the  inflectional  tangent  at 
this  point  intersects  four  consecutive  generators  of  our  surface. 
On  each  generator  there  will  be  in  general  two  flecnode  points, 

'  W.,  pp.  102,  104,  112. 

*  A.  Cayley,  Collected  Mathematical  Papers,  II,  29. 

J  Cambridge  and  Dublin  Mathematical  Journal,  IV  (1849),  252-60. 


INTRODUCTION  3 

since  four  consecutive  generators  of  a  ruled  surface  have  two 
straight-line  intersectors.  As  x  varies  these  two  points  trace  a 
flecnode  curve  of  two  branches.  The  tangents  to  the  asymptotic 
curves  through  the  flecnode  points  generate  a  flecnode  surface  of 
two  sheets.  For  any  system  of  form  (i)  the  flecnodes  are  obtained 
by  factoring  the  quadratic  covariant 

C  =  Ui2Z^ —U2iy^-\-  {uij,—U22)yz. 

When  a  ruled  surface  5  is  referred  to  its  flecnode  curves,  W12  =  U21  =  o. 
We  may  without  loss  of  generality  take  ^„  =  P22  =  o.  Under  these 
conditions  the  system  of  differential  equations  for  one  sheet  F'-^''  of 
the  flecnode  surface  may  be  written'^ 


(9) 


Pi2  P12 


p"-\-[2(q,^-^q22)-pi2p2i]y'-2^  p'+    2q',,-p,2q2i-4T^qii  \y 

Pl2  I  Pl2        J 


—  gjaP  =  0 


where 

p  =  2y'+pi2Z,        (x  =  2z'+p2iy. 

In  precisely  the  same  way  the  system  for  the  second  sheet  F<~'^ 
may  be  written,  if  in  (9)  we  transpose  the  subscripts  and  replace 
y  and  p  by  z  and  a,  respectively.  This  system  we  shall  denote 
by  (10). 

We  shall  call  /^^'^  the  first  flecnode  transform  of  S.  We  shall  call 
F'-~^^  the  minus  first  flecnode  transform  of  S  for  the  reason  that^ 
the  first  flecnode  transform  of  F'-~''^  is  S.  The  minus  first  transform 
of  F^'^  is  S.  The  first  transform  of  F^'^  we  shall  call  the  second  trans- 
form of  5  and  denote  it  by  F'-^K  Continuing  in  this  way  we  obtain 
a  suite  of  surfaces  which  we  shall  call  the  flecnode  suite.  Questions 
naturally  arise  as  to  the  cases  in  which  this  suite  either  terminates 
or  returns  again  into  itself.  This  paper  concerns  itself  with  some 
of  these  questions. 

The  author  wishes  to  express  to  Professor  Wilczynski  his  deep 
appreciation  for  his  genuine  interest  and  helpful  criticism. 

'  W.,  p.  153.  =  IMd.,  p.  178. 


I.     CASES  OF  TERMINATION 

A  flecnode  tangent  is  not  ordinarily  tangent  to  the  flecnode 
curve.  If  it  were,  since  it  is  at  the  same  time  tangent  to  the 
asymptotic  curve  which  passes  through  the  flecnode  point,  the 
flecnode  curve  would  be  a  straight  line.  The  corresponding  sheet 
of  the  flecnode  surface  then  degenerates  into  a  straight  Une.  In 
this  event  any  ruled  surface  made  up  of  the  Unes  intersecting 
this  line  may  be  called  its  flecnode  surface.  If  we  try  to  continue 
the  flecnode  suite,  we  are  powerless  to  determine  which  of  these 
surfaces  to  select.  We  shall  say  in  this  case  that  the  flecnode 
suite  terminates  with  its  first  transform,  since  it  cannot  be  con- 
tinued in  unambiguous  fashion.  A  necessary  and  sufficient  condition 
that  the  flecnode  suite  terminate  with  its  first  transform  is  dio  =  o, 
i.e.,  the  given  ruled  surface  has  a  straight-line  directrix. 

Since  the  ruled  surface  has  a  straight-Hne  directrix  let  us  take 
this  line  as  one  of  the  reference  curves  Cy.  Let  C^  be  an  arbitrary 
curve.     We  may  write 

^^^^  \Zi=/i(^),       Zj=/j(x),      Zj=fj{x),       Z4=I. 

Let  us  compute  the  system  (i)  for  which  these  are  the  funda- 
mental solutions.     We  obtain^  the  following  system, 

z''+[xf','-f--x^-^+f^)y'-^^z'^ 
whence  by  (6) 

««=2j/3,:«:K/»-^/o+2(:»^'+2)(/r--^')-3;p-Wx''-/n 

•h2(xfr-fn, 

'  W.,  p.  128. 

4 


CASES  OF  TERMINATION 


where  \f3,x\  indicates  the  Schwarzian  derivative  of /j  with  respect 
to  X.    The  flecnodes  are  given  by 

(12)  V  =  yy       f=-«2i>'-«222- 


(13) 


The  second  flecnode  point  J"  is  given  by 

Si^  W21     ^22/1 ) 

Sa^  U21X      W22/2  > 

S  3  ~  W22/3  , 

f4=  ^22  • 


We  obtain  the  following  result:  The  equations  of  the  most  general 
curve  which  can  serve  as  the  second  branch  of  the  flecnode  curve  of  a 
ruled  surface  with  a  straight-line  directrix  can  he  obtained  without  any 
integration. 

If  now  we  make  upon  y  and  z  the  transformation  indicated  by 
(12)  we  refer  our  surface  to  its  flecnode  curves.  The  resulting 
equivalent  system  of  form  (i)  is 


(14) 


77" =0, 

f"+(  W2l-2^^^^'+/>22W2I-/'2lW22   ]n 
U21U22 


+  <H 


-2U2,- 2W^^— +^22«2I-/'22— ]V 

Uo<2  W22  *^22  Uo 


1      I   W22  ,      W22  U2i   .  . 

+  ( P22—  —  2-J-   ]i=0. 


Carpenter  has  shown^  that  when  one  branch  of  the  flecnode 
curve  is  rectilinear  the  second  branch  of  the  flecnode  curve  may 
be  plane.  In  this  case  he  has  shown  that  p2i  can  be  determined 
except  for  two  constants  of  integration.  He  has  found,  moreover, 
that  the  second  branch  of  the  flecnode  curve  may  be  a  conic, 
in  which  special  case  the  equations  of  form  (i)  take  the  form 


(15) 


y  =0, 
z"-\-icy-iz=o. 


We  have  found  that  d^o  =  o  is  a  condition  for  the  termination  of 
the  flecnode  suite  with  its  first  transform.    Let  us  suppose  0io5^o 

'  A.  F.  Carpenter,  "Ruled  Surfaces  Whose  Flecnode  Curves  Have  Plane  Branches," 
Transactions  of  the  American  Mathematical  Society,  XVI  (1915),  529,  hereafter  referred 
to  as  C. 


FLECNODE  TRANSFORMATION  OF  RULED  SURFACES 


and  that  the  suite  terminates  with  the  second  transform.  We 
shall  then  have  ^io  =  o.  We  desire  to  express  this  condition  in 
terms  of  the  invariants  of  the  original  form  (i). 

The  differential  equations  for  F'-^^  are  given  by  (9),  but  in  a  form 
which  is  not  convenient  for  the  computation  of  its  invariants. 
We  proceed  to  transform  (9)  into  an  equivalent  system  for  which 
Pil  =  p2l  =  Uii  =  U2l  =  o.  To  do  this  we  first  refer  the  surface  to 
its  flecnode  curves.  Furthermore,  we  shall  specialize  the  inde- 
pendent variable  so  as  to  make  w  =  i.  To  refer  the  surface  to  its 
flecnode  curves  we  put 


(16) 

The  resulting  system  is 


y=y,     Y=-^y+p. 


■y-_.^y_r-(g-i|+,..),+|-r=o, 


(17) 


+ 


2!^-#+2(9„+9,,)-^,/>.,]y 


-( 


L  P 

'K. 

.Pl2 

Ip'l 


p 
Pl^p 


iP'A   pi. 


f.  ^*t:-S('"+'">- 


Pl2P'21 


4 


■f  2gxib' 


+9.JF  =  o. 


By  the  choice  of  suitable  multiphers  we  may  transform  (17)  into  a 
system  which  preserves  the  condition  Wj"  =  w"j  =  o  and  satisfies 
the  further  condition  pfi=pf2  =  o.    We  accomplish  this  by  putting 


(18) 


y=p.2v,    Y=Y, 


where  />i2  7«^o  on  account  of  the  assumption  ^wt^o,  a  transformation 
which  reduces  (17)  to 


P 


,+^K=o, 


(19)   { 


Y"+  \2p'A-^+2p,Mr,+q,,)  -p\2p2^  T,' 

Ap'^2-^+h§+pUq^^+q22)-pr2p2.p'u-hP\2pn+2q[,p,Ari 

Pl2  Pl2  J 


11 
2PI2 


-\-q22)Y  =  o. 


CASES  OF  TERMINATION  7 

We  are  now  in  a  position  to  compute  conveniently  the  invariants  of 
F^^\    In  particular  we  have^ 

where  we  use  the  upper  index  i  systematically  for  the  quantities 
referring  to  the  surface  /^"'.    We  find 

(20)  e=-2^^-§-2(9„+9,,)+^./»... 

Pl2         P12 

Since  d^o  is  an  invariant  of  the  system  (17)  which  is  geometrically 
determined  by  (i),  it  must  also  be  an  invariant  of  (i).     We  have^ 


(21) 


P12       I 

Pl2         lOPIo 

Pl2         4''I0 


Substituting  these  values  in  (20)  we  obtain 

(22)        C'  =  ^(-i60x'o-8^;^xo+4^9C-^,+  i2ei2+04-i)- 

I  of  10 

In  order  to  put  ^"o  into  a  form  in  which  its  invariant  character 
will  be  apparent  we  seek  to  replace  the  derivatives  occurring  in 
(22)  by  invariants  and  to  put  into  evidence  the  isobaric  property 
which  has  been  masked  by  the  assumption  ^4=1.     We  find  that 

C  =  -^(3^S-2M,o-20|^A+20|Mx4-^4^9+O4.i)- 

is  an  invariant  which  reduces  to  (22)  under  the  assumption  ^4  =  1. 
Consequently, 

^3O  =  3^S-2Mao-2e|eA+20fMi4-^4^9+O4-i  =  O 

is  a  necessary  and  sufficient  condition  for  theflecnode  suite  to  terminate 
with  its  second  transform. 

One  naturally  inquires  about  the  case  d[^''^  =  o.     It  is  obtained 
from  the  foregoing  case  ^"0  =  0  by  merely  changing  the  sign  of  d\. 

'W.,p.  119.  '  Ibid.,  p.  120. 


8         FLECNODE  TRANSFORMATION  OF  RULED  SURFACES 

If  ^30  =  0,  we  may  use  the  methods  of  the  first  part  of  this 
section  to  compute  the  second  branch  of  the  flecnode  curve  on  F*". 
This  curve  is  at  the  same  time  a  branch  of  the  flecnode  curve  on  5. 
We  can  obtain  both  branches  of  the  flecnode  curve  on  S,  by  apply- 
ing the  minus  first  flecnode  transformation  to  F"'.  Thus  we  see 
that  the  eqtiations  of  both  branches  of  the  flecnode  curve  on  S  may 
be  determined  without  any  integration  if  the  flecnode  suite  terminates 
with  its  second  transform. 


II.     CASES  IN  WHICH  THE  GENERATORS  OF  THE 

SECOND  AND  MINUS  SECOND 

TRANSFORMS  INTERSECT 

Let  us  inquire  whether  the  generators  of  the  second  and  minus 
second  flecnode  transforms  can  intersect.     Consider  the  quantities 


(23) 


pki)  =  2y'  —  2^  y  —  p  , 


II    =2p'-{-[2(q,^-hq22)-pi2p2i]y-2^p, 


formed  from  (9)  just  as  p  and  cr  were  formed  from  (i).  Since  the 
original  system  has  been  taken  in  the  form  for  which  pji  =  P22  =  W12  = 
U2x  =  o,  we  have 

2qiI=Pl2  ,         2/>i2/>2I— 4(911+922)  =«Ii  +  «22  , 

pi2Z  =  p—2y',  2p'+^i2<r  =  Wii3'- 

If  we  substitute  these  in  (23)  we  find 


(24) 


,(!)  = 


P12 


y-px2z, 
P'. 


^uy——-p—pi2(r 


Every  point  on  the  line  p"V  has  the  property^  that  the  straight 
Une  joining  it  to  the  corresponding  point  on  the  generator  of  F"' 
is  an  asymptotic  tangent  of  i^"\  This  correspondence  is  expressed* 
by  the  fact  that  the  corresponding  variables  undergo  cogredient 
transformation.  In  particular,  the  line  a^^^Y  will  be  the  tangent 
to  the  asymptotic  curve  through  the  second  flecnode  point,  and 
consequently  a  generator  of  F^^\  if,  and  only  if,  o-"'  is  that  point 
on  the  line  p"V  corresponding  to  F  as  a  point  on  the  Une  yp.  But 
we  have 

F  =  <}'-w«P, 


'W.,  p.  146. 


'  Loc.  cit. 


lo       FLECNODE  TRANSFORMATION  OF  RULED  SURFACES 
and  therefore 

Using  (24)  we  find 

(25)  (rW  =  (-2  ^ <-|«mW  )y-p^,uSz-\-2  ^  «(')p+/;,.MW(r . 

\  Pl^  I  Pl2 

Any  point  on  the  hne  (r*"F  is  given  by  the  expression 

(26)  [(-2  ^^  <-^«w^^Aa+</3l>;-/>,,<az+  (2  ^  w«-7^W^V 

Similarly  any  point  on  the  line  a^~'^^Z  is  given  by 


(27) 


— /).-i/(-« 


/'2i«^r"73'+ 


l«^(-i)_2?£i«(-»)y+ 


«ir"5]i 


+/>,,«( -I)  7P+  /a  ^  m(-^)7-m(-')5  W , 


where  the  upper  index  —  i  refers  to  the  system  (10).  The  point 
Pz  is  the  second  flecnode  point  on  F^~'\  and  o-^~*^  is  the  point  on 
the  line  p^~^V  corresponding  to  it.  The  quantities  p^~'^  and  v  are 
obtained  from  (10)  just  as  p  and  <t  are  from  (i). 

In  order  that  a  point  on  the  hne  given  by  (26)  may  also  be  on 
the  Hne  given  by  (27),  we  must  have 


(28) 


P12 
/'i2«(''a+*-2  ^  m'-'>co7+w(-"cj5  =  o  , 

p2l 


where  co  is  a  proportionaUty  factor.  This  requires  the  vanishing 
of  the  determinant  of  the  system,  which  after  a  combining  of 
columns  can  be  written  in  the  form 


CASES  IN  WHICH  GENERATORS  INTERSECT 


II 


(29) 


or 

(30) 


2 
O 


^21  P21U21 


PI2U 


(I) 


o 


2 

o  w^-') 


4pi2p2i(ulT'^u^'^-u^^^u'-'^y-{uu^-'^u^^^y  =  o 


We  proceed  to  express  (30)  in  terms  of  the  invariants  of  S. 
We  have  u  =  u^~^''  =  u'-^K    Our  condition  (30)  can  be  written 


(31) 


Now  we  have^ 


4pi2p2i{ui^  "  +  M^iO'  — W4  =  0  ' 


U'  =  i 


4q=-u, 


P12     P'2: 


BaOq 


\Pl2         p2l/  4^10  ' 

M.=|,   »;r»+«Si'=«(g-|;). 

Substituting  these  values  in  (31)  we  find 
(32)  t?i8=^9-^Ao=o. 

In  the  foregoing  we  have  assumed  d^r^o,  6 ^0^0,  cases  dis- 
cussed elsewhere  in  this  paper. 

If  ??i8  =  o,  we  may  recover  again  the  determinant  of  the  system 
(28)  whose  vanishing  is  a  necessary  and  sufficient  condition  that  the 
system  have  a  non-trivial  solution.  The  quantities  a,  (3,  C07,  co5  are, 
in  fact,  proportional  to  the  cofactors  of  the  elements  of  any  row  in 
the  determinant  of  the  coefficients.  Since  the  cofactors  of  the 
corresponding  elements  in  the  rows  are  proportional  we  are  led  to  a 
unique  set  of  ratios  unless  all  of  these  cofactors  vanish,  i.e.,  we 
have  a  definite  point  of  intersection.  Consequently,  ??i8  =  o  is  a 
necessary  and  sufficient  condition  for  the  generators  of  the  second  and 
minus  second  transforms  to  intersect. 

'  W.,  p.  119. 


12       FLECNODE  TRANSFORMATION  OF  RULED  SURFACES 

If  we  choose  the  independent  variable  so  that  ^4  =  1,  the  condi- 
tion (30)  becomes 

(33)  4(Pl2p21-p21p'uy-pl2p2l  =  0. 

We  may  solve  this  relation  in  symmetric  fashion  by  putting 

P21 
Then  {;^^)  gives  upon  integration 

(34)  Q=cJ^. 

If  we  assume  pn  =  p22  =  Ui2  =  U2j.  =  u—i=o  (32)  becomes 

(35)  ^-^10=0. 

Under  the  same  assumptions  Carpenter^  has  shown  that  Cy  is  a 
conic  if,  and  only  if, 

(36)  e,+2e',o=o. 

We  shall  assume  d^y^o,  for  if  ^9  =  0  then  by  (35)  0io  =  o,  a  case  dis- 
cussed elsewhere  in  this  paper.     Differentiating  (35)  we  have 

(37)  '  26,6^-6^0=0. 

If  now  we  eliminate  d[o  from  (36)  and  (37)  we  find,  since  6^9'^ o, 

(38)  4^;+i  =  o. 

Integrating  (38)  we  get 

69=-i(x-\-c). 
From  (35)  we  find 

6^o=Mx■^cy. 

Using  the  condition^  that  Cy  be  a  plane  curve  we  have 

Moreover,   we   have   assumed    ^4=1.     Thus   we   have   obtained 
expressions  for  the  four  fundamental  invariants  in  terms  of  the 

'C,  p.  515.  '  Ibid.,  p.  S16. 


CASES  IN  WHICH  GENERATORS  INTERSECT  13 

independent  variable  and  one  arbitrary  constant.  We  may  now 
compute^  explicitly  the  coefficients  of  (i).  They  lead  us  to  the 
system 

(y"-hdz'-iy  =  o, 

(39)  I  2-+ j^(:,+,)y+_I_(^_,)3,=o , 

where  c  and  d  are  two  arbitrary  constants.  However,  one  of  these 
constants  is  not  essential  since 

y  =  ay,    z=bz,     ^  =  x-\-l, 

the  most  general  transformation  which  leaves  our  conditions  undis- 
turbed, serves  when 

^=d^  ^=''  ^=r 

to  remove  d.     The  resulting  system  is 

(y"+kz'-\y=o, 

I  16        16 

For  every  value  of  k,  (40)  defines  a  class  of  mutually  projective 
ruled  surfaces.  We  see  then  that  there  exists  a  single  infinity  of 
classes  of  mutually  projective  ruled  surfaces  the  generators  of  whose 
second  and  minus  second  flecnode  transforms  intersect,  which  have 
the  additional  property  that  the  flecnode  curve  consists  of  two  distinct 
branches,  one  of  which  is  a  conic.  It  can  readily  be  shown  by  a 
direct  test  that  the  system  (40)  has  all  of  the  properties  attributed 
to  it. 

Let  us  inquire  whether  in  addition  to  the  foregoing  the  second 
branch  of  the  flecnode  curve  may  be  plane.  The  necessary  condi- 
tion is^ 

(41)  2Mio-3^io+^o=0. 

But  the  values  of  d^o  determined  above  clearly  do  not  satisfy  (41). 
So  we  conclude  that  there  are  no  ruled  surfaces  whose  second  and 
minus  second  flecnode  generators  intersect,  which  have  the  property 

'  W.,  p.  120.  2  c,  p.  516.   ■ 


14       FLECNODE  TRANSFORMATION  OF  RULED  SURFACES 


that  their  flecnode  curve  consists  of  two  distinct  plane  branches,  one 
of  which  is  a  conic. 

We  proceed  to  compute  the  invariants  of  F^'^  and  F^~^^  in  terms 
of  the  invariants  of  S  for  the  case  t?i8  =  o.  For  this  purpose  we  shall 
use  the  form  (19)  in  which  the  surface  is  referred  to  its  flecnode 
curve  and  multipliers  for  the  dependent  variables  have  been  chosen 
so  as  to  remove  certain  derivatives.  The  equations  for  F'-'^^  may 
be  obtained  from  (19)  by  transposing  the  subscripts  provided  the 
dependent  variables  f  and  Z  are  put  in  place  of  77  and  Y.  We 
find  the  following  values  for  the  invariants  of  system  (19) 

L     F«       /*"  •  J 


(42) 


P12         Pl2 

e^:i=i6(-2^+s§+P^2P»). 

\  Pz2  Pa 


The  corresponding  invariants  for  F^~'^  may  be  obtained  from  these 
by  transposing  the  subscripts.     If  t?i8  =  o,  we  have 


[  p^2p^^=\{2e',o+e^) ,  p2xpa=li2d',o-e,) , 


(43) 


911+922=^(16^10—^4.1) , 


P^2  16^, 


(8^10^10+2^9010+^10  — 4^10)  > 


l^  =  -7V(32Oxo'-480ioOi'o-I209^io+l609Mio+24^i'o+^9U, 
Pu        04tfJo 


giving  the  following  values  for  0"',  dg\  etc.,  in  terms  of  the  invari- 
ants of  S,  upon  the  hypothesis  that  the  independent  variable  is 
chosen  so  as  to  make  ^4=1, 

^9"  =  ^(-32^o^x'o'+48Mx'oC+I209^io-l6Mxo0i';-240x? 


(44) 


-  ^9^10+  20x0^4-1+  ^9^0^4-1+  2^o^;x)  , 


^"0  =-^(-l6Mx';+I20.'?-0xo+O4.x)  , 
10^0 

^i'J=^(-i6Mii+8e,0,'o+.exo+2O0iS+i6^o) 


CASES  IN  WHICH  GENERATORS  INTERSECT  15 

For  the  computation  of  the  invariants  of  F'-~^'>  the  equations  (43) 
remain  vaUd  if  we  transpose  the  subscripts  in  the  left  members  and 
replace  6^  by  —  dg  in  the  right  members.  Consequently,  equations 
(44)  with  the  latter  change  give  us  the  corresponding  invariants  of 

If  we  abandon  the  hypothesis  64  =  1,  we  find  the  following 

/.--)         J  —^4  ^9^10 ~ ^0^15^4-1+ ^4^9^10^4-1  ~^o^is)  > 


I 


where^ 


-10  z-/)2"(~  2^10^20+3^5 ""^^10+^0^4.1)  , 

lOtflo 

C  =  i-(-2M.o-4^!^As+^5^io+5^s) , 


^IS  —  5^io^4~  204^10  ,  ^15  =  5^4.1^4—2^4^4.1  , 


To  verify  (45)  it  is  sufficient  to  note  that  these  formulae  reduce  to 

(44)  when  ^4  =  1,  and  that  the  right  members  of  (45)  are  invariants 
of  5. 

The  invariants  of  F'-~'^^  are  given  by  a  system  obtained  from 

(45)  by  replacing  dg  by  -6^. 

Since  0io  =  0i7"  we  conclude  that  in  case  the  generators  of  the 
second  and  minus  second  flecnode  transforms  intersect,  the  minus 
first  flecnode  surface  belongs  to  a  special  linear  complex  if,  and  only 
if,  the  first  flecnode  surface  belongs  to  a  special  linear  complex.  In 
this  event  the  second  and  minus  second  flecnode  surfaces  are  straight 
lines  having  a  point  in  common. 

Let  us  inquire  whether  the  corresponding  absolute  invariants 
of  F'-^^  and  F^~^^  may  not  have  the  same  values.  Equations  (44) 
and  the  corresponding  ones  for  F''~^'>  show  that  it  will  then  be 
necessary  that 

'  W.,  p.  112. 


[  1209^10—  16^9^10^10  — ^9^io~l~^9^o^4-i  =  0  • 


i6       FLECNODE  TRANSFORMATION  OF  RULED  SURFACES 

We  shall  again  assume  that  6g9^o.  Since  t?i8  =  o  we  must  then 
have  also  Bio9^o.    The  equations  (46)  reduce  to 

(47)  ^10=0,    04.1=1. 

Equations  (44)  show  that  in  this  case 

so  we  conclude  that  the  absolute  invariants  of  the  first  and  minus 
first  flecnode  transforms  are  equM  if,  and  only  if,  these  surfaces  belong 
to  special  linear  complexes. 


III.    CASES  OF  PERIODICITY 

It  has  been  shown^  that  each  sheet  of  the  flecnode  surface  of  S 
has  S  itself  as  one  of  the  sheets  of  its  flecnode  surface.  Thus 
the  minus  first  transform  of  7^'"  is  S.  The  first  transform  will 
ordinarily  be  a  new  surface  F^'\  Let  us  inquire  whether  it  too 
can  coincide  with  S.  If  so,  the  flecnode  suite  will  be  periodic,  of 
period  two.  In  this  event  the  two  sheets  of  the  flecnode  surface 
of  F^"  must  coincide.  Then  the  two  sheets  of  the  flecnode  surface 
of  5  cannot  be  distinct,  else'  they  would  be  distinct  on  /^'".  Thus 
we  see  that  ^4  =  0  is  a  necessary  condition.  It  is  evident  geo- 
metrically that  this  condition  is  also  sufficient.  We  conclude  then 
that  the  flecnode  suite  is  periodic,  of  period  two,  when,  and  only 
when,  the  flecnode  curve  meets  every  generator  in  two  coincident  points, 
or  is  indeterminate. 

This  theorem  may  also  be  established  analytically.  The 
ruled  surfaces  for  which  the  flecnode  curves  are  indeterminate  are 
quadrics.  In  this  case  we  have  not  only  ^4  =  0,  but  Wi2  =  W2i  =  Wii  — 
U22  —  0.  The  flecnode  surface  in  this  case  coincides  with  the  original 
quadric  generated  by  the  generators  of  the  second  kind.  The 
second  flecnode  surface  is  the  original  surface  generated  again  by 
its  first  set  of  rulings. 

We  may  now  assume  6 ^9^0  and  ^lor^o  since  the  cases  excluded 
by  this  hypothesis  have  been  considered  already.  The  generators 
of  the  first  and  minus  first  transforms  are^  generators  of  the  second 
kind  on  the  quadric  which  osculates  our  original  ruled  surface  along 
a  generator.  They  are  distinct  since  we  have  assumed  dj^9^o. 
Consequently,  they  cannot  intersect  and,  therefore,  the  flecnode 
suite  cannot  be  periodic,  of  period  three. 

This  theorem,  too,  may  be  established  analytically.  We  pro- 
ceed to  determine  whether  the  flecnode  suite  can  be  of  period 
four.  We  shall  assume  again  d^j^o  and  ^io?^o.  We  require  that 
the  line  joining  the  point  Y  to  the  point  Z  be  a  generator  of  F^'\ 

'  W.,  p.  178.  'Ibid.,  p.  147. 

17 


1 8       FLECNODE  TRANSFORMATION  OF  RULED  SURFACES 

In  particular,  it  is  necessary  that  the  point  (r"\  given  by  (25),  shall 
be  a  point  on  this  Une.  Now,  any  point  on  the  line  YZ  is  given 
by  the  expression 

Consequently,  we  must  have 


(48) 


M2'a+    2 


-<+^MW^*Mw  =  0, 


P12 


Since  the  three  quantities  a,  ^,  00  cannot  all  be  zero,  it  is  necessary 
that  the  second  order  determinants 


(49) 


P12U21 


< 

Pl2 

} 

«(^> 

Pl2 

shall  both  vanish.  Now  pi2^o,  else  the  hypothesis  ^10 5^0  is  con- 
tradicted. Moreover,  w"Vo,  since  the  flecnode  curves  are  dis- 
tinct on  S,  and  are  therefore  distinct  also  on  F^".  It  is  necessary 
then  that 


(50) 


M^r'V^)-w'VM^-^>=M=o, 


However,  w  =  o  implies  ^4  =  0,  contrary  to  hypothesis.  So  we 
must  conclude  that  the  flecnode  suite  cannot  he  periodic,  of  period 
four. 


VITA 

John  Wayne  Lasley,  Jr.,  was  born  in  Burlington,  North  CaroHna,  Sep- 
tember 2  2,  1 89 1.  After  attending  the  public  schools  of  his  native  town  he 
entered  the  University  of  North  Carolina  in  1906,  graduating  with  the  degree 
of  Bachelor  of  Arts  in  1910.  He  received  the  degree  of  Master  of  Arts  from 
this  institution  in  191 1.  In  191 5-16  he  attended  the  Johns  Hopkins  Uni- 
versity, studying  with  Professors  Bateman,  Coble,  Cohen,  and  Morley.  Dur- 
ing the  summers  of  1917,  191 8,  191 9,  and  the  year  1919-20,  he  was  in  residence 
at  the  University  of  Chicago,  studying  with  Professors  Birkhoff,  BUss,  Dickson, 
Moore,  and  Wilczynski.  Since  receiving  his  Bachelor's  degree  during  the 
years  not  specified  above  he  has  taught  in  the  University  of  North  CaroUna  in 
the  capacity  of  instructor,  assistant  professor,  and,  associate  professor.  In 
connection  with  this  work  he  studied  with  Professors  Cain  and  Henderson. 
To  all  of  his  teachers  he  is  indebted  for  inspiration  and  instruction.  To 
Professor  Wilczynski  he  is  particularly  gratefxil  for  sympathetic  interest  and 
helpful  advice  during  the  preparation  of  this  dissertation. 


19 


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